We prove the finite generation of canonical rings of smooth projective varieties of general type defined over complex numbers. MSC 32J25

In this paper we partially answer two open questions concerning clean rings. First, we demonstrate that if a quasi-continuous module is strongly clean then it is Dedekind-finite. Second, we prove a partial converse. We also prove that all clean decompositions on submodules of continuous modules extend to the entire module.

A graph property is a set of (countable) graphs. A homomorphism from a graph G to a graphH is an edge-preserving map from the vertex set of G into the vertex set of H; if such a map exists, we write G → H. Given any graph H, the hom-property →H is the set of H-colourable graphs, i.e., the set of all graphs G satisfying G → H. A graph property P is of finite character if, whenever we have that F...

Nakayama (Ann. of Math. 42, 1941) showed that over an artinian serial ring every module is a direct sum of uniserial modules. Hence artinian serial rings have the property that each right (left) ideal is a finite direct sum of quasi-injective right (left) ideals. A ring with the property that each right (left) ideal is a finite direct sum of quasi-injective right (left) ideals will be called a ...

This paper presents a coding theorem for linear coding over finite rings, in the setting of the Slepian–Wolf source coding problem. This theorem covers corresponding achievability theorems of Elias (IRE Conv. Rec. 1955, 3, 37–46) and Csiszár (IEEE Trans. Inf. Theory 1982, 28, 585–592) for linear coding over finite fields as special cases. In addition, it is shown that, for any set of finite cor...

We study reversible polymerization of rings. In this stochastic process, two monomers bond and, as a consequence, two disjoint rings may merge into a compound ring or a single ring may split into two fragment rings. This aggregation-fragmentation process exhibits a percolation transition with a finite-ring phase in which all rings have microscopic length and a giant-ring phase where macroscopic...

Abstract We obtain a nontrivial bound on the number of solutions to equation $$ \begin{align*} &\sum_{i=1}^{\nu} A^{x_i} = \sum_{i=\nu+1}^{2\nu} A^{x_i}, \qquad 1 \leqslant x_i \tau, \end{align*}$$with fixed $n\times n$ matrix $A$ over finite field ${{\mathbb {F}}}_q$ $q$ elements multiplicative order $\tau $. apply our result new for additive character sums with exponential function, beyon...